`(p to q) wedge (q to -p)` is equivalent to

Options

A. 1. q
B. 2. `~q`
C. 3. p
D. 4. `~p`

Answer

To solve the problem `(p to q) wedge (q to -p)`, we need to analyze the logical expressions involved. Let's break it down step by step.

### Step 1: Understand the symbols
- **p to q** (denoted as \( p \implies q \)): This is a logical implication that is false only when \( p \) is true and \( q \) is false.
- **q to -p** (denoted as \( q \implies \neg p \)): This is also a logical implication that is false only when \( q \) is true and \( \neg p \) (not p) is false, which means \( p \) is true.

### Step 2: Construct the truth table
We will create a truth table for \( p \), \( q \), \( p \implies q \), \( \neg p \), and \( q \implies \neg p \).

| p | q | \( p \implies q \) | \( \neg p \) | \( q \implies \neg p \) |
|-------|-------|---------------------|---------------|--------------------------|
| T | T | T | F | F |
| T | F | F | F | F |
| F | T | T | T | T |
| F | F | T | T | T |

### Step 3: Evaluate \( (p \implies q) \wedge (q \implies \neg p) \)
Now we will evaluate the conjunction of \( p \implies q \) and \( q \implies \neg p \).

| p | q | \( p \implies q \) | \( q \implies \neg p \) | \( (p \implies q) \wedge (q \implies \neg p) \) |
|-------|-------|---------------------|--------------------------|--------------------------------------------------|
| T | T | T | F | F |
| T | F | F | F | F |
| F | T | T | T | T |
| F | F | T | T | T |

### Step 4: Analyze the results
From the truth table for \( (p \implies q) \wedge (q \implies \neg p) \), we see that the expression is true when:
- \( p = F \) and \( q = T \)
- \( p = F \) and \( q = F \)

This corresponds to the truth values of \( \neg p \):
- When \( p = F \), \( \neg p = T \).
- When \( p = T \), \( \neg p = F \).

### Step 5: Conclusion
The final truth table for \( (p \implies q) \wedge (q \implies \neg p) \) matches the truth table for \( \neg p \). Therefore, we conclude that:

**The expression \( (p \implies q) \wedge (q \implies \neg p) \) is equivalent to \( \neg p \).**

### Final Answer
The equivalent expression is \( \neg p \).

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